Propositional Logic · step 11 of 13

Converse, inverse & contrapositive

From the conditional p → q you can form three relatives: the converse q → p, the inverse ¬p → ¬q, and the contrapositive ¬q → ¬p. Only the contrapositive is logically equivalent to the original — and the converse and inverse are equivalent to each other. This single fact powers proof by contraposition, and confusing the converse with the original is one of the most common reasoning errors in and out of mathematics.

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Method: how to approach it

The order below is what examiners expect to see, and each step carries its own marks.

Identify the antecedent and consequentRewrite the sentence in clean "if p then q" form first. English orderings such as "q, provided p" hide which part is which.
Converse — swapq → p. Swap the two parts and leave the negations alone.
Inverse — negate¬p → ¬q. Keep the order and negate both parts.
Contrapositive — swap and negate¬q → ¬p. Do both operations. This is the one you can safely substitute for the original.

Worked example

For "If a number is divisible by 6, then it is divisible by 3", state the converse, inverse and contrapositive, and say which are true.

Let p = "n is divisible by 6", q = "n is divisible by 3". The original p → q is true.
Converse q → p: "If n is divisible by 3 then it is divisible by 6" — false, e.g. n = 9.
Inverse ¬p → ¬q: "If n is not divisible by 6 then it is not divisible by 3" — false, same witness n = 9.
Contrapositive ¬q → ¬p: "If n is not divisible by 3 then it is not divisible by 6" — true.

Answer. Original and contrapositive are true; converse and inverse are both false, witnessed by n = 9.

Where marks get dropped

These are the specific errors that cost credit on converse, inverse & contrapositive questions — QED's rubric penalises each of them separately.

Assuming the converse follows from the original. It does not, and the example above shows why — this fallacy has its own name, affirming the consequent.
Negating without swapping and calling it the contrapositive. That is the inverse, which is not equivalent.
Forgetting that the converse and inverse are contrapositives of each other, so they always share a truth value. If you find one is false, the other is too.

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Converse, inverse & contrapositive — frequently asked questions

Why is proof by contraposition valid?

Because ¬q → ¬p is logically equivalent to p → q — the two have identical truth tables. So proving one proves the other, and sometimes the negated version is far easier to work with.

If a statement and its converse are both true, what do I have?

A biconditional, p ↔ q. That is exactly what "if and only if" asserts, which is why iff proofs are always done in two directions.

Is the contrapositive the same as proof by contradiction?

No, though they are often confused. Contraposition proves ¬q → ¬p directly. Contradiction assumes p ∧ ¬q and derives an absurdity. Both are valid, but the structure and the marks differ.

The rest of Propositional Logic

Connectives, truth tables, equivalences and normal forms. Each subtopic below has its own method, worked example and mark-losing traps.

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